hpfilter: Hodrick-Prescott filter of a time series
In mFilter: Miscellaneous Time Series Filters

Description Usage Arguments Details Value Author(s) References See Also Examples

This function implements the Hodrick-Prescott for estimating cyclical and trend component of a time series. The function computes cyclical and trend components of the time series using a frequency cut-off or smoothness parameter.

1	hpfilter(x,freq=NULL,type=c("lambda","frequency"),drift=FALSE)

`x`	a regular time series.
`type`	character, indicating the filter type, `"lambda"`, for the filter that uses smoothness penalty parameter of the Hodrick-Prescott filter (default), `"frequency"`, for the filter that uses a frequency cut-off type Hodrick-Prescott filter. These are related by lambda = (2sin(pi/frequency))^{-4}*.
`freq`	integer, if `type="lambda"` then `freq` is the smoothing parameter (lambda) of the Hodrick-Prescott filter, if `type="frequency"` then `freq` is the cut-off frequency of the Hodrick-Prescott filter.
`drift`	logical, `FALSE` if no drift in time series (default), `TRUE` if drift in time series.

Almost all filters in this package can be put into the following framework. Given a time series \{x_t\}^T_{t=1} we are interested in isolating component of x_t, denoted y_t with period of oscillations between p_l and p_u, where 2 ≤ p_l < p_u < ∞.

Consider the following decomposition of the time series

x_t = y_t + \bar{x}_t

The component y_t is assumed to have power only in the frequencies in the interval \{(a,b) \cup (-a,-b)\} \in (-π, π). a and b are related to p_l and p_u by

a=\frac{2 π}{p_u}\ \ \ \ \ {b=\frac{2 π}{p_l}}

If infinite amount of data is available, then we can use the ideal bandpass filter

y_t = B(L)x_t

where the filter, B(L), is given in terms of the lag operator L and defined as

B(L) = ∑^∞_{j=-∞} B_j L^j, \ \ \ L^k x_t = x_{t-k}

The ideal bandpass filter weights are given by

B_j = \frac{\sin(jb)-\sin(ja)}{π j}

B_0=\frac{b-a}{π}

The Hodrick-Prescott filter obtains the filter weights \hat{B}_j as a solution to

\hat{B}_{j}= \arg \min E \{ (y_t-\hat{y}_t)^2 \} = \arg \min ≤ft\{ ∑^{T}_{t=1}(y_t-\hat{y}_{t})^2 + λ∑^{T-1}_{t=2}(\hat{y}_{t+1}-2\hat{y}_{t}+\hat{y}_{t-1})^2 \right\}

The Hodrick-Prescott filter is a finite data approximation with following moving average weights

\hat{B}_j=\frac{1}{2π}\int^{π}_{-π} \frac{4λ(1-\cos(ω))^2}{1+4λ(1-\cos(ω))^2}e^{i ω j} d ω

If drift=TRUE the drift adjusted series is obtained as

\tilde{x}_{t}=x_t-t≤ft(\frac{x_{T}-x_{1}}{T-1}\right), \ \ t=0,1,…,T-1

where \tilde{x}_{t} is the undrifted series.

A "mFilter" object (see mFilter).

Mehmet Balcilar, mehmet@mbalcilar.net

M. Baxter and R.G. King. Measuring business cycles: Approximate bandpass filters. The Review of Economics and Statistics, 81(4):575-93, 1999.

L. Christiano and T.J. Fitzgerald. The bandpass filter. International Economic Review, 44(2):435-65, 2003.

J. D. Hamilton. Time series analysis. Princeton, 1994.

R.J. Hodrick and E.C. Prescott. Postwar US business cycles: an empirical investigation. Journal of Money, Credit, and Banking, 29(1):1-16, 1997.

R.G. King and S.T. Rebelo. Low frequency filtering and real business cycles. Journal of Economic Dynamics and Control, 17(1-2):207-31, 1993.

D.S.G. Pollock. Trend estimation and de-trending via rational square-wave filters. Journal of Econometrics, 99:317-334, 2000.

mFilter, bwfilter, cffilter, bkfilter, trfilter

## library(mFilter)

data(unemp)

opar <- par(no.readonly=TRUE)

unemp.hp <- hpfilter(unemp)
plot(unemp.hp)
unemp.hp1 <- hpfilter(unemp, drift=TRUE)
unemp.hp2 <- hpfilter(unemp, freq=800, drift=TRUE)
unemp.hp3 <- hpfilter(unemp, freq=12,type="frequency",drift=TRUE)
unemp.hp4 <- hpfilter(unemp, freq=52,type="frequency",drift=TRUE)

par(mfrow=c(2,1),mar=c(3,3,2,1),cex=.8)
plot(unemp.hp1$x,  ylim=c(2,13),
main="Hodrick-Prescott filter of unemployment: Trend, drift=TRUE",
     col=1, ylab="")
lines(unemp.hp1$trend,col=2)
lines(unemp.hp2$trend,col=3)
lines(unemp.hp3$trend,col=4)
lines(unemp.hp4$trend,col=5)
legend("topleft",legend=c("series", "lambda=1600", "lambda=800",
       "freq=12", "freq=52"), col=1:5, lty=rep(1,5), ncol=1)

plot(unemp.hp1$cycle,
main="Hodrick-Prescott filter of unemployment: Cycle,drift=TRUE",
     col=2, ylab="", ylim=range(unemp.hp4$cycle,na.rm=TRUE))
lines(unemp.hp2$cycle,col=3)
lines(unemp.hp3$cycle,col=4)
lines(unemp.hp4$cycle,col=5)
## legend("topleft",legend=c("lambda=1600", "lambda=800",
## "freq=12", "freq=52"), col=1:5, lty=rep(1,5), ncol=1)

par(opar)